arXiv subjects

Yuki Satake

Publications and source records attributed to Yuki Satake.

4 recordsLinked to original sources

arXiv2026

Factorized Krylov subspace methods for solving large Sylvester equations

Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments.

math.NA↗

arXiv2023

Matrix equation representation of the convolution equation and its unique solvability

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

math.NA↗

arXiv2020

Program Verification via Predicate Constraint Satisfiability Modulo Theories

This paper presents a verification framework based on a new class of predicate Constraint Satisfaction Problems called pCSP where constraints are represented as clauses modulo first-order theories over function variables and predicate variables that may represent well-founded predicates. The verification framework generalizes an existing one based on Constrained Horn Clauses (CHCs) to arbitrary clauses, function variables, and well-foundedness constraints. While it is known that the satisfiability of CHCs and the validity of queries for Constrained Logic Programs (CLP) are inter-reducible, we show that, thanks to the added expressiveness, pCSP is expressive enough to express muCLP queries. muCLP itself is a new extension of CLP that we propose in this paper. It extends CLP with arbitrarily nested inductive and co-inductive predicates and is equi-expressive as first-order fixpoint logic. We show that muCLP can naturally encode a wide variety of verification problems including but not limited to termination/non-termination verification and even full modal mu-calculus model checking of programs written in various languages. To establish our verification framework, we present (1) a sound and complete reduction algorithm from muCLP to pCSP and (2) a constraint solving method for pCSP based on stratified CounterExample-Guided Inductive Synthesis (CEGIS) of (co-)inductive invariants, ranking functions, and Skolem functions witnessing existential quantifiers. Stratified CEGIS combines CEGIS with stratified families of templates to achieve relative completeness and faster and stable convergence of CEGIS by avoiding the overfitting problem. We have implemented the proposed framework and obtained promising results on diverse verification problems that are beyond the scope of the previous verification frameworks based on CHCs.

cs.PL↗

arXiv2019

Relation between the T-congruence Sylvester equation and the generalized Sylvester equation

The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, where $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times m}$ are given, and $X\in\mathbb{R}^{n\times m}$ is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices $A$ and $B$ are assumed to be square matrices ($m=n$). In this paper, two transformations are provided for rectangular matrices $A$ and $B$. One of them is an extension of the result of Oozawa et al. for the case $m\ge n$, and the other is a novel transformation for the case $m\le n$.

math.NA↗